1. Field of the Invention
The present invention relates to a method of gradual deformation of representations or realizations, generated by sequential simulation, of a model of a heterogeneous medium which is not limited to a Gaussian stochastic model, based on a gradual deformation algorithm of Gaussian stochastic models.
2. Description of the Prior Art
In French patent application 98/09,018 a method is described which gradually deforms a stochastic (Gaussian type or similar) model of a heterogeneous medium such as an underground zone, constrained by a set of parameters relative to the structure of the medium. This method comprises drawing a number p (p=2. for example) of realizations (or representations) independent of the model or of at least part of the selected model of the medium from all the possible realizations and one or more iterative stages of gradual deformation of the model by performing one or more successive linear combinations of p independent initial realizations and then composite realizations are successively obtained possibly with new draws, etc., the coefficients of this combination being such that the sum of their squares is 1.
Gaussian or similar models are well-suited for modelling continuous quantity fields but are therefore ill-suited for modelling zones crossed by fracture networks or channel systems for example.
The most commonly used geostatistical simulation algorithms are those referred to as sequential simulation algorithms. Although sequential simulation algorithms are particularly well-suited for simulation of Gaussian models, these algorithms are not in principle limited to this type of model.
A geostatistical representation of an underground zone is formed for example by subdividing thereof by a network with N meshes and by determining a random vector with N dimensions Z=(Z1, Z2, . . . ZN) best corresponding to measurements or observations obtained on the zone. As shown for example by Johnson, M. E.; in xe2x80x9cMultivariate Statistical Simulationxe2x80x9d; Wiley and Sons, New York, 1987, this approach reduces the problem of the creation of an N-dimensional vector to a series of N one-dimensional problems. Such a random vector is neither necessarily multi-Gaussian nor stationary. Sequential simulation of Z first involves the definition of an order according to which the N elements (Z1, Z2, . . . ZN) of vector Z are generated one after the other. Apart from any particular case, it is assumed that the N elements of Z are generated in sequence from Z1 to ZN. To determine a value of each element Z1, (i=1, . . . , N), the following operations have to be carried out:
a) building the distribution of Zi conditioned by (Z1, Z2. . . Zi-1) Fc(Zi)=P(Zixe2x89xa6/Z1, Z2, . . . Zi-1); and
b) determining a value of Zi from distribution Fc (Zi).
In geostatistical practice, sequential simulation is frequently used to generate multi-Gaussian vectors and non-Gaussian indicator vectors. The main function of sequential simulation is to determine conditional distributions Fc (Zi) (i=1, . . . , N). Algorithms and softwares for estimating these distributions are for example described in:
Deutsch, C. V. et al. xe2x80x9cGSLIB (Geostatistical Software Library) and Useres Guidexe2x80x9d; Oxford University Press, New York, Oxford 1992.
Conceming determining the values from distribution Fc (Zi), there also is a wide set of known algorithms.
The inverse distribution method is considered by means of which a realization of Zi; i=Fcxe2x88x921(ul) is obtained, where u1 is taken from a uniform distribution between 0 and 1. A realization of vector Z therefore corresponds to a realization of vector U whose elements U1, U2, . . . , UN, are mutually independent and evenly distributed between 0 and 1.
It can be seen that a sequential simulation is an operation S which converts a uniform vector U=(U1, U2, . . . UN) to a structured vector Z=(Z1, Z2, . . . , ZN):
Z=S(U)xe2x80x83xe2x80x83(1).
The problem of the constraint of a vector Z to various types of data can be solved by constraining conditional distributions Fc(Zi) (i=1, . . . , N) and/or uniform vector U=(U1, U2, . . . , UN).
Recent work on the sequential algorithm was focused on improving the estimation of conditional distributions Fc (zi) by geologic data and seismic data integration. An article by Zhu, H. et al: xe2x80x9cFormatting and Integrating Soft Data: Stochastic imaging via the Markov-Bayes Algorithmxe2x80x9d in Soares, A., Ed. GeostaUstics Troia 92, vol.l: KiuwerAcad. Pubi., Dordrecht, The Netherlands, pp.1-12, 1993 is an example.
However, this approach cannot be extended to integration of non-linear data such as pressures from well tests and production records, unless a severe linearization is imposed. Furthermore, since any combination of uniform vectors U does not give a uniform vector, the method for gradual deformation of a stochastic model developed in the aforementioned patent application cannot be directly applied within the scope of the sequential technique described above.
The method according to the invention thus allows making the two approaches compatible, that is to extend the formalism developed in the aforementioned patent application to gradual deformation of realizations, generated by sequential simulation, of a model which is not limited to a Gaussian stochastic model.
The method allows gradual deformation of a representation or realization, generated by sequential simulation, of a model which is not limited to a Gaussian stochastic model of a physical quantity z in a heterogeneous medium such as an underground zone, in order to constrain the model to a set of data collected in the medium by previous measurements and observations relative to the state or the structure thereof.
The method of the invention comprises applying an algorithm of gradual deformation of a stochastic model to a Gaussian vector (Y) having a number N of mutually independent variables that is connected to a uniform vector (U) with N mutually independent uniform variables by a Gaussian distribution function (G), so as to define a chain of realizations u(t) of vector (U), and using these realizations u(t) to generate realizations z(t) of this physical quantity that are adjusted in relation to the (non-linear) data.
According to a first embodiment, the chain of realizations u(t) of uniform vector (U) is defined from a linear combination of realizations of Gaussian vector (Y) whose combination coefficients are such that the sum of their squares is one.
According to another embodiment, gradual deformation of a number n of parts of the model representative of the heterogeneous model is performed while preserving the continuity between these n parts of the model by subdividing uniform vector (U) into a number n of mutually independent subvectors.
The method of the invention finds applications in modeling underground zones which generates representations showing how a certain physical quantity is distributed in an underground zone (permeability z for example) and which is compatible in the best manner with observed or measured data: geologic data, seismic records, measurements obtained in wells, notably measurements of the variation with time of the pressure and of the flow rate of fluids from a reservoir, etc.